## Tangents and Normals

A tangent or normal to a curve is a line, taking the form where is the gradient and is the intercept. Given a function we can find the gradient at by finding the gradient function and substituting the value into this expression. Sometimes however we don't have so is not given explicitly as a function of In these cases typically we have to differentiate implicitly and find as a function of both and and then substitute a point into the expression for to find the gradient at that point. Finally substitute into the equation to find the equation of the line.

Example: Find the equation of the tangent to the curve at the point We differentiate implicitly to get The gradient at the point is  Example: Find the equation of the tangent to the curve at the point We differentiate implicitly to get We have to make the subject of this equation. The gradient at the point is  Example: Find the equation of the normal to the curve at the point We differentiate implicitly to get We have to make the subject of this equation. The gradient at the point is   